Why is it that the covariant derivative of a covariant tensor does not seem to follow the product rule like contravariant tensors do when taking the covariant derivatives of those? Here is a visual of what I mean: This is the covariant derivative of a contravariant vector. As you can see, it perfectly follows the product rule. ∇nVm = ∂Vm/∂yn + [itex]\Gamma[/itex]mnrVr where the Cristoffel symbol is simply a mixed derivative. Where as with a covariant vector: ∇bVa = ∂Va/∂yb - [itex]\Gamma[/itex]cbaVc As you can see with the covariant vector, all of the terms are the same ones that appear if you use the product rule when differentiating the covariant transformation of the vector with respect to b. However, that minus sign would not appear when you use the product rule. Can anyone explain to me then, why the covariant derivatives of covariant tensors like this have that minus sign in the middle instead of a plus sign (which would go in accordance with the product rule)?